Nuprl Lemma : implies-qle
∀[a,b:ℚ]. a ≤ b supposing ∀n:ℕ+. (a ≤ (b + (1/n)))
Proof
Definitions occuring in Statement :
qle: r ≤ s,
qdiv: (r/s),
qadd: r + s,
rationals: ℚ,
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
iff: P ⇐⇒ Q,
true: True,
guard: {T},
qsub: r - s,
squash: ↓T,
rev_uimplies: rev_uimplies(P;Q),
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
qless_complement_qorder,
qless_wf,
qle_witness,
nat_plus_wf,
qle_wf,
qadd_wf,
qdiv_wf,
subtype_rel_set,
rationals_wf,
less_than_wf,
istype-int,
int-subtype-rationals,
nat_plus_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
set_subtype_base,
int_subtype_base,
iff_weakening_uiff,
equal-wf-base,
int-equal-in-rationals,
qsub_wf,
qmul_wf,
uiff_transitivity2,
qadd_preserves_qless,
squash_wf,
true_wf,
qmul_over_plus_qrng,
qmul_zero_qrng,
qinv_inv_q,
mon_assoc_q,
qadd_comm_q,
qadd_ac_1_q,
qinverse_q,
mon_ident_q,
qless_transitivity_2_qorder,
qle_weakening_eq_qorder,
qless_irreflexivity,
q-archimedean,
qless-int,
nat_properties,
decidable__lt,
intformnot_wf,
itermAdd_wf,
intformle_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
le_wf,
qadd-add,
istype-less_than,
qmul_preserves_qle,
qmul-qdiv-cancel,
subtype_rel_self,
iff_weakening_equal,
qmul_one_qrng,
qmul_comm_qrng,
qmul_over_minus_qrng,
qadd_preserves_qle,
qadd_inv_assoc_q
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
independent_isectElimination,
lambdaFormation_alt,
universeIsType,
hypothesis,
independent_functionElimination,
sqequalRule,
functionIsType,
natural_numberEquality,
applyEquality,
because_Cache,
intEquality,
lambdaEquality_alt,
setElimination,
rename,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
equalityIstype,
baseClosed,
sqequalBase,
equalitySymmetry,
inhabitedIsType,
isectIsTypeImplies,
minusEquality,
imageElimination,
equalityTransitivity,
imageMemberEquality,
addEquality,
unionElimination,
baseApply,
closedConclusion,
dependent_set_memberEquality_alt,
instantiate,
universeEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[a,b:\mBbbQ{}]. a \mleq{} b supposing \mforall{}n:\mBbbN{}\msupplus{}. (a \mleq{} (b + (1/n)))
Date html generated:
2019_10_16-PM-00_37_03
Last ObjectModification:
2019_06_26-PM-05_06_49
Theory : rationals
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